I am talking about a relation that is what Wikipedia describes as left-unique and right-unique. I never heard these terms before, but I have heard of the alternatives (injective and functional). The question is, which terminology do you recommend? Should I include short definitions? (The context is a text in the area of formal methods. I'm not sure if this helps.)

These are some trade-offs that I see:

I think that left-unique and right-unique are not widely known, but I'm not sure at all.

functional is overloaded

injective sounds too fancy (subjective, of course)

left-unique and right-unique are symmetric (good, of course)

Edit: It seems the question is unclear. Here are more details. I describe sets X and Y and then say:

now we must find an injective and functional relation between sets X and Y such that...

now we must find a left-unique and right unique relation between sets X and Y...

Which one do you recommend? What other information would you add? The relation does not have to be total. For example, various different ranges correspond to different 'feasible' relations. Technically I should not need to say that the relation does not have to be total, but will many people assume that it has to be total if I don't say it?

I certainly did not know the terms left-unique and right-unique, and moreover when I tried to guess what they meant, I ended up with the opposite meanings. Left-unique, I reasoned, must mean that a pair in the relation is uniquely determined by its left member, i.e., functional. But that is the definition of right-unique. Go figure
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Harald Hanche-OlsenMar 11 '10 at 13:28

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People in formal methods know the standard usages, which are "injective" and "functional". If you're worried about "functional" being taken to mean "higher-order function", then use the phrase "functional relation", as in "$R$ is a functional relation".
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Neel KrishnaswamiMar 11 '10 at 14:21

@Harald: I guessed the same way as you did. @Neel: Thanks. At the moment I'm inclined to say we must find an injective and functional relation between X and Y, and not define injective/functional.
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rgrigMar 11 '10 at 17:43

1 Answer
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Injective and functional are completely standard in this case. This is what you should use. The term "functional" is not overloaded, when you are using it to say that something is a function. Being functional means exactly that the relation is a function.

A relation that is injective and functional is precisely an injective function on its domain. It is a bijection of its domain with its range.

If you don't want to think of the relation as a function, then you can also describe it as a one-to-one correspondence of its domain with its range.

(And I don't think any of these terms I suggest would need to be defined, since their meaning is fairly universally known. This would definitely not be true of left-unique and right-unique.)

Joel, I think most people take R is a function from X to Y to mean that for each $x\in X$ there is exactly one $y\in Y$ such that $xRy$; similarly, I think most people take R is a bijection between X and Y to mean that R and its inverse are both functions. Also, I think most people use one-to-one correspondence as a synonym for bijection. That is not what I want to say. What I want to say is that for each $x\in X$ there is at most one $y\in Y$ such that $xRy$ and vice-versa. (This is what the left-unique and right-unique definitions that I pointed to say.)
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rgrigMar 11 '10 at 17:38

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Well, I never said R should be a function from X to Y, or a bijection between X and Y, but rather, that it is a function on its domain, or a bijection of its domain with its range. The domain of a relation R is the set of x for which there is y with xRy, and the range is the corresponding set of y. These may not be X and Y, respectively, and this should resolve the confusion. It is completely correct to say that a relation is functional if and only if it is a function from its domain to its range, and this is why the word functional is used.
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Joel David HamkinsMar 11 '10 at 18:45

In particular, what I mean to say is that I stand by my answer.
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Joel David HamkinsMar 11 '10 at 18:55

@Joel: I agree. I never said you were wrong. I am just pointing out that neither 'bijection' nor 'one-to-one correspondence' mean the same as (left-unique and right-unique), so they aren't what I need to say. (I did up-vote your answer :), just so you know.)
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rgrigMar 11 '10 at 19:27

Rgrig, but if you say "bijection of its domain with its range" or "one-to-one correspondence of its domain with its range", as I suggested, then it IS what you mean to say.
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Joel David HamkinsMar 11 '10 at 19:46